Properties

Label 5054.2.a.bj
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 15 x^{7} + 40 x^{6} + 81 x^{5} - 162 x^{4} - 205 x^{3} + 204 x^{2} + 210 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta_{1} q^{3} + q^{4} + \beta_{6} q^{5} + \beta_{1} q^{6} - q^{7} - q^{8} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q - q^{2} -\beta_{1} q^{3} + q^{4} + \beta_{6} q^{5} + \beta_{1} q^{6} - q^{7} - q^{8} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{9} -\beta_{6} q^{10} + ( -1 - 2 \beta_{3} + \beta_{6} + \beta_{8} ) q^{11} -\beta_{1} q^{12} + ( -1 - \beta_{1} - \beta_{2} - \beta_{8} ) q^{13} + q^{14} + ( 2 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{15} + q^{16} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{17} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{18} + \beta_{6} q^{20} + \beta_{1} q^{21} + ( 1 + 2 \beta_{3} - \beta_{6} - \beta_{8} ) q^{22} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{23} + \beta_{1} q^{24} + ( -\beta_{3} + \beta_{6} - \beta_{8} ) q^{25} + ( 1 + \beta_{1} + \beta_{2} + \beta_{8} ) q^{26} + ( -2 - \beta_{3} - \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{27} - q^{28} + ( \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{29} + ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{30} + ( -1 + 2 \beta_{5} - \beta_{6} + \beta_{8} ) q^{31} - q^{32} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{33} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{34} -\beta_{6} q^{35} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{36} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{8} ) q^{37} + ( 3 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{39} -\beta_{6} q^{40} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} + 3 \beta_{6} + \beta_{8} ) q^{41} -\beta_{1} q^{42} + ( 4 - 3 \beta_{1} + \beta_{2} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{43} + ( -1 - 2 \beta_{3} + \beta_{6} + \beta_{8} ) q^{44} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{45} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{46} + ( 3 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{47} -\beta_{1} q^{48} + q^{49} + ( \beta_{3} - \beta_{6} + \beta_{8} ) q^{50} + ( -4 - 3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} ) q^{51} + ( -1 - \beta_{1} - \beta_{2} - \beta_{8} ) q^{52} + ( -1 - \beta_{1} - \beta_{2} + 4 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{53} + ( 2 + \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{54} + ( 3 + \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{8} ) q^{55} + q^{56} + ( -\beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{58} + ( -1 - \beta_{1} - 5 \beta_{3} - \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{59} + ( 2 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{60} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{61} + ( 1 - 2 \beta_{5} + \beta_{6} - \beta_{8} ) q^{62} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{63} + q^{64} + ( 1 - \beta_{3} - 3 \beta_{5} - 2 \beta_{7} ) q^{65} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{66} + ( -3 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{67} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{68} + ( -1 + \beta_{2} + 5 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{69} + \beta_{6} q^{70} + ( -2 + \beta_{1} - \beta_{2} - 4 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{71} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{72} + ( 5 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{73} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{8} ) q^{74} + ( \beta_{1} + \beta_{2} + 5 \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{8} ) q^{75} + ( 1 + 2 \beta_{3} - \beta_{6} - \beta_{8} ) q^{77} + ( -3 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{78} + ( -3 + \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{6} - \beta_{8} ) q^{79} + \beta_{6} q^{80} + ( -2 + \beta_{1} - \beta_{2} + 4 \beta_{3} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{81} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{6} - \beta_{8} ) q^{82} + ( \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + 3 \beta_{8} ) q^{83} + \beta_{1} q^{84} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{4} + 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{85} + ( -4 + 3 \beta_{1} - \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{86} + ( 2 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{7} ) q^{87} + ( 1 + 2 \beta_{3} - \beta_{6} - \beta_{8} ) q^{88} + ( 3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{89} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{90} + ( 1 + \beta_{1} + \beta_{2} + \beta_{8} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{92} + ( -1 - \beta_{2} - 5 \beta_{3} - \beta_{4} - 4 \beta_{5} + 3 \beta_{7} ) q^{93} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{94} + \beta_{1} q^{96} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{97} - q^{98} + ( -2 - \beta_{1} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{2} - 3q^{3} + 9q^{4} + 3q^{5} + 3q^{6} - 9q^{7} - 9q^{8} + 12q^{9} + O(q^{10}) \) \( 9q - 9q^{2} - 3q^{3} + 9q^{4} + 3q^{5} + 3q^{6} - 9q^{7} - 9q^{8} + 12q^{9} - 3q^{10} - 3q^{11} - 3q^{12} - 12q^{13} + 9q^{14} + 6q^{15} + 9q^{16} + 12q^{17} - 12q^{18} + 3q^{20} + 3q^{21} + 3q^{22} + 6q^{23} + 3q^{24} + 12q^{26} - 24q^{27} - 9q^{28} + 6q^{29} - 6q^{30} - 9q^{31} - 9q^{32} + 3q^{33} - 12q^{34} - 3q^{35} + 12q^{36} - 9q^{37} + 33q^{39} - 3q^{40} - 3q^{41} - 3q^{42} + 21q^{43} - 3q^{44} + 18q^{45} - 6q^{46} + 21q^{47} - 3q^{48} + 9q^{49} - 39q^{51} - 12q^{52} + 24q^{54} + 24q^{55} + 9q^{56} - 6q^{58} + 9q^{59} + 6q^{60} + 6q^{61} + 9q^{62} - 12q^{63} + 9q^{64} + 3q^{65} - 3q^{66} - 27q^{67} + 12q^{68} - 6q^{69} + 3q^{70} - 9q^{71} - 12q^{72} + 51q^{73} + 9q^{74} + 3q^{75} + 3q^{77} - 33q^{78} - 24q^{79} + 3q^{80} - 3q^{81} + 3q^{82} + 9q^{83} + 3q^{84} + 6q^{85} - 21q^{86} - 3q^{87} + 3q^{88} - 9q^{89} - 18q^{90} + 12q^{91} + 6q^{92} + 3q^{93} - 21q^{94} + 3q^{96} + 12q^{97} - 9q^{98} - 15q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 15 x^{7} + 40 x^{6} + 81 x^{5} - 162 x^{4} - 205 x^{3} + 204 x^{2} + 210 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{8} + 9 \nu^{7} - 280 \nu^{6} + 463 \nu^{5} + 2443 \nu^{4} - 4613 \nu^{3} - 4884 \nu^{2} + 7433 \nu + 1878 \)\()/1156\)
\(\beta_{3}\)\(=\)\((\)\( 9 \nu^{8} + 27 \nu^{7} - 262 \nu^{6} - 345 \nu^{5} + 2127 \nu^{4} + 1189 \nu^{3} - 5404 \nu^{2} - 1399 \nu + 2166 \)\()/1156\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{8} - 27 \nu^{7} + 262 \nu^{6} + 345 \nu^{5} - 2127 \nu^{4} - 1189 \nu^{3} + 6560 \nu^{2} + 243 \nu - 6790 \)\()/1156\)
\(\beta_{5}\)\(=\)\((\)\( -32 \nu^{8} + 193 \nu^{7} + 193 \nu^{6} - 2434 \nu^{5} + 433 \nu^{4} + 9227 \nu^{3} - 1369 \nu^{2} - 11274 \nu - 1825 \)\()/1156\)
\(\beta_{6}\)\(=\)\((\)\( -43 \nu^{8} + 160 \nu^{7} + 449 \nu^{6} - 1627 \nu^{5} - 1396 \nu^{4} + 3214 \nu^{3} + 2667 \nu^{2} + 1161 \nu - 1197 \)\()/1156\)
\(\beta_{7}\)\(=\)\((\)\( -97 \nu^{8} + 287 \nu^{7} + 1154 \nu^{6} - 3025 \nu^{5} - 4043 \nu^{4} + 7929 \nu^{3} + 4746 \nu^{2} - 4895 \nu - 32 \)\()/1156\)
\(\beta_{8}\)\(=\)\((\)\( 151 \nu^{8} - 414 \nu^{7} - 1859 \nu^{6} + 4423 \nu^{5} + 6690 \nu^{4} - 11488 \nu^{3} - 7981 \nu^{2} + 5171 \nu + 1179 \)\()/1156\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + 6 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{8} + 3 \beta_{7} - \beta_{6} - 2 \beta_{5} + 9 \beta_{4} + 13 \beta_{3} - \beta_{2} + 10 \beta_{1} + 25\)
\(\nu^{5}\)\(=\)\(14 \beta_{8} + 27 \beta_{7} - 8 \beta_{6} - 4 \beta_{5} + 14 \beta_{4} + 18 \beta_{3} - \beta_{2} + 46 \beta_{1} + 22\)
\(\nu^{6}\)\(=\)\(25 \beta_{8} + 56 \beta_{7} - 7 \beta_{6} - 30 \beta_{5} + 81 \beta_{4} + 131 \beta_{3} - 18 \beta_{2} + 97 \beta_{1} + 181\)
\(\nu^{7}\)\(=\)\(167 \beta_{8} + 309 \beta_{7} - 45 \beta_{6} - 68 \beta_{5} + 160 \beta_{4} + 242 \beta_{3} - 31 \beta_{2} + 402 \beta_{1} + 221\)
\(\nu^{8}\)\(=\)\(395 \beta_{8} + 765 \beta_{7} - 7 \beta_{6} - 350 \beta_{5} + 756 \beta_{4} + 1302 \beta_{3} - 233 \beta_{2} + 981 \beta_{1} + 1438\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.30594
2.67565
2.18756
1.68026
−0.00478425
−0.906887
−1.47653
−1.81276
−2.64845
−1.00000 −3.30594 1.00000 1.60566 3.30594 −1.00000 −1.00000 7.92924 −1.60566
1.2 −1.00000 −2.67565 1.00000 −2.78796 2.67565 −1.00000 −1.00000 4.15909 2.78796
1.3 −1.00000 −2.18756 1.00000 −0.616139 2.18756 −1.00000 −1.00000 1.78540 0.616139
1.4 −1.00000 −1.68026 1.00000 3.48998 1.68026 −1.00000 −1.00000 −0.176713 −3.48998
1.5 −1.00000 0.00478425 1.00000 −1.04022 −0.00478425 −1.00000 −1.00000 −2.99998 1.04022
1.6 −1.00000 0.906887 1.00000 −1.94678 −0.906887 −1.00000 −1.00000 −2.17756 1.94678
1.7 −1.00000 1.47653 1.00000 −1.23411 −1.47653 −1.00000 −1.00000 −0.819871 1.23411
1.8 −1.00000 1.81276 1.00000 3.21562 −1.81276 −1.00000 −1.00000 0.286090 −3.21562
1.9 −1.00000 2.64845 1.00000 2.31395 −2.64845 −1.00000 −1.00000 4.01430 −2.31395
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5054.2.a.bj 9
19.b odd 2 1 5054.2.a.bk 9
19.f odd 18 2 266.2.u.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.u.c 18 19.f odd 18 2
5054.2.a.bj 9 1.a even 1 1 trivial
5054.2.a.bk 9 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\(T_{3}^{9} + \cdots\)
\(T_{5}^{9} - \cdots\)
\(T_{13}^{9} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{9} \)
$3$ \( -1 + 210 T - 204 T^{2} - 205 T^{3} + 162 T^{4} + 81 T^{5} - 40 T^{6} - 15 T^{7} + 3 T^{8} + T^{9} \)
$5$ \( 179 + 468 T + 132 T^{2} - 393 T^{3} - 171 T^{4} + 123 T^{5} + 43 T^{6} - 18 T^{7} - 3 T^{8} + T^{9} \)
$7$ \( ( 1 + T )^{9} \)
$11$ \( 8704 + 4608 T - 9216 T^{2} - 4192 T^{3} + 1800 T^{4} + 756 T^{5} - 129 T^{6} - 48 T^{7} + 3 T^{8} + T^{9} \)
$13$ \( -10099 - 9801 T + 21333 T^{2} + 19388 T^{3} + 2796 T^{4} - 1491 T^{5} - 451 T^{6} + 3 T^{7} + 12 T^{8} + T^{9} \)
$17$ \( -1216 + 6912 T - 9072 T^{2} + 1808 T^{3} + 3360 T^{4} - 2280 T^{5} + 477 T^{6} + 3 T^{7} - 12 T^{8} + T^{9} \)
$19$ \( T^{9} \)
$23$ \( 49096 - 107472 T + 64890 T^{2} + 1117 T^{3} - 10296 T^{4} + 1473 T^{5} + 446 T^{6} - 78 T^{7} - 6 T^{8} + T^{9} \)
$29$ \( 23104 + 120384 T + 82032 T^{2} - 85392 T^{3} - 23064 T^{4} + 6516 T^{5} + 797 T^{6} - 153 T^{7} - 6 T^{8} + T^{9} \)
$31$ \( -239104 - 286464 T - 70464 T^{2} + 33760 T^{3} + 15936 T^{4} - 96 T^{5} - 785 T^{6} - 66 T^{7} + 9 T^{8} + T^{9} \)
$37$ \( -2078144 - 3602112 T - 763296 T^{2} + 350768 T^{3} + 130476 T^{4} + 5580 T^{5} - 2123 T^{6} - 186 T^{7} + 9 T^{8} + T^{9} \)
$41$ \( -12718144 + 10258752 T + 964992 T^{2} - 1043008 T^{3} + 11556 T^{4} + 28476 T^{5} - 553 T^{6} - 294 T^{7} + 3 T^{8} + T^{9} \)
$43$ \( 17294912 - 10187328 T - 177888 T^{2} + 1022800 T^{3} - 146988 T^{4} - 14040 T^{5} + 3625 T^{6} - 57 T^{7} - 21 T^{8} + T^{9} \)
$47$ \( 5202496 - 4737408 T + 421824 T^{2} + 432528 T^{3} - 79500 T^{4} - 8736 T^{5} + 2441 T^{6} - 12 T^{7} - 21 T^{8} + T^{9} \)
$53$ \( -56512 + 80640 T + 80496 T^{2} - 103632 T^{3} - 22356 T^{4} + 11088 T^{5} + 249 T^{6} - 207 T^{7} + T^{9} \)
$59$ \( -83156571 + 39966048 T + 15118272 T^{2} - 2809467 T^{3} - 415395 T^{4} + 56637 T^{5} + 3465 T^{6} - 414 T^{7} - 9 T^{8} + T^{9} \)
$61$ \( 4158872 - 1162224 T - 3511542 T^{2} - 1072183 T^{3} + 7272 T^{4} + 29397 T^{5} + 884 T^{6} - 288 T^{7} - 6 T^{8} + T^{9} \)
$67$ \( 2331584 + 3670848 T + 1646544 T^{2} - 83584 T^{3} - 242988 T^{4} - 62916 T^{5} - 5541 T^{6} + 24 T^{7} + 27 T^{8} + T^{9} \)
$71$ \( 703 + 27024 T - 70692 T^{2} - 41341 T^{3} + 60009 T^{4} + 3759 T^{5} - 2313 T^{6} - 216 T^{7} + 9 T^{8} + T^{9} \)
$73$ \( -212405696 + 127313280 T - 20485008 T^{2} - 1793728 T^{3} + 851388 T^{4} - 68016 T^{5} - 3771 T^{6} + 894 T^{7} - 51 T^{8} + T^{9} \)
$79$ \( 243181 + 6095523 T + 5453835 T^{2} + 1592564 T^{3} + 84699 T^{4} - 36432 T^{5} - 5547 T^{6} - 72 T^{7} + 24 T^{8} + T^{9} \)
$83$ \( 9838979 - 42896586 T + 12822210 T^{2} + 603039 T^{3} - 516159 T^{4} + 29157 T^{5} + 4519 T^{6} - 384 T^{7} - 9 T^{8} + T^{9} \)
$89$ \( -43964992 - 72767616 T - 27029376 T^{2} - 1660336 T^{3} + 550020 T^{4} + 55692 T^{5} - 3765 T^{6} - 429 T^{7} + 9 T^{8} + T^{9} \)
$97$ \( 11829248 - 7938048 T - 4313088 T^{2} + 4038144 T^{3} - 797184 T^{4} + 24192 T^{5} + 6304 T^{6} - 408 T^{7} - 12 T^{8} + T^{9} \)
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